The nature of Claire Messud's novels – the elegance of their construction and prose, the wry, precise sentences, her powerfully accurate descriptions – makes it somehow unlikely that she'd tell me about a twee sign in a bakery near where she lives in Cambridge, Massachusetts. "Life is not about waiting for the storm to pass," the sign reads, "it's about learning to dance in the rain." It just doesn't seem very Messud in either sentiment or delivery, and indeed she remarks that it's the kind of thing she would once have "scoffed at". But we have been talking about middle age (she is 46), and about how its consolations – a certain kind of relaxation with oneself, or an improvement in material circumstances – are offset by such harrowing milestones as parental illness and death. Once those kinds of experience have entered your life, adages start to seem slightly less objectionable on mere aesthetic grounds; as Messud puts it, "super-tacky, but not wrong".

Messud uses the verb "scoff" at one other point, when she recalls a first-person piece she read in a magazine, some years ago, while killing time in a waiting room. It was about a woman who had only just realised that you weren't graded at the end of your life. At first it seemed a ridiculous notion but as she read on, it began to dawn on Messud that she half-identified with it. Now, she says, "I still believe on some level that at the end somebody will say: you get an A-minus for your life. And it's not true. It's not true."

These reflections about the passing of time, the way that a life is reckoned, come in the context of our discussion about Messud's latest novel, The Woman Upstairs, which has prompted, in both its critical reception and more general feature coverage, a great deal of heat (and a little less light). It concerns a Massachusetts elementary school teacher and frustrated artist of 42 called Nora Eldridge, an unmarried and childless woman who, we gradually discover, has spent much of her adult life elevating duty and decorum over artistic and emotional freedom – she has been, in her own shorthand, the perfect "woman upstairs", cheerful, responsible, undemanding. And she was carrying it off pretty well until an unexpected passion threw her off course. The novel opens with these words: "How angry am I? You don't want to know. Nobody wants to know about that." Its second paragraph concludes with this pronouncement: "It was supposed to say 'Great Artist' on my tombstone, but if I died right now it would say 'such a good teacher/daughter/friend' instead; and what I really want to shout, and want in big letters on that grave, too, is FUCK YOU ALL."

It is an arresting, combative, in-your-face opening, which Messud explains came to her before the rest of the novel, discrete and compact "like a marble". The following pages fill with a complex story, subtle and humorous and poignant. It's a story of what happens when love arrives in your life not in the conveniently parcelled form of one other, single, available person, but in the shape of a family, each of whom sparks a different variety of desire and attachment. The Shahids – a visiting academic, his wife and young son – are lovable to Nora in and of themselves, but they also represent a wide range of possibilities; of intellectual, artistic, sensual, erotic and even maternal fulfilment, of the potential for delight. And yet Nora's anger – and the consequent issue of her "likability" – is what has formed the basis of much of the conversation about the book.

If that focus has made Messud herself angry, she doesn't show it here; it puts her in mind, she says, of a character in Thomas Bernhard's novel The Loser – "His characters are great ranters," she chuckles – who rails against cultural shrinkage with the words: "All of Kant gets reduced to one little shrunken head!" She understands why it happens: "We all want to feel we've comprehended something, fully grasped it, and that involves reducing it." Nonetheless, she would certainly like to correct so one-dimensional a reading – or, at least, clarify it. "People say, 'Oh, it's a book about an angry woman,' and I say, 'Well she's angry at the beginning, which is also the end, but her anger is actually a response to how joyful and full of wonder and discovery and hope this time was in her life, and what she feels that she's at risk of losing or may have lost.' I feel I do have to remind people, or else they think it's just 200 pages of yelling."

She didn't want to convey a message, she insists; more than anything else, she wanted to "create a person" in her entirety, and to explore an interior life. "The moment that I had in mind in a way was in Chekhov's "The Lady with the Dog". He's having this affair with the lady, and he's walking his child to school, and he thinks, 'How can it be that the thing that's most important to me in my whole life, nobody knows about but me?' It's not shared with anybody. And then he thinks, 'That's true for everyone else, too.' It's a moment of apprehension of the state of the world; all of us are going around with an entire story of our lives, completely different from the story of our lives that anybody else would tell. So much of our lives never breaks the surface."

But capturing a whole subjectivity – both its surface manifestation and the mysteries of its interior – involves confronting realities that are not always palatable. To those who have found Nora hard to like, Messud points out that she would seem extremely likable if you met her, "and that's the point. It costs her a great deal to be a likable character … underneath that she feels she has had to hide, up to this point, her real thoughts, her real dreams, her real desires, because they would be in some way unseemly or perhaps offputting to people." And yet some readers have asked Messud whether Nora – who is by definition unreliable because she's a first-person narrator, whose story is partial because she's the only character we hear from – is insane.

A counter-reading, though, might suggest that she is too sane; so determined to keep everything on an even keel that she has sacrificed the kind of spontaneity and wildness necessary to fall in love, or to make art. A key question the novel's title invites us to consider is to what extent this has befallen her because she's a woman. "Is that a gendered state?" says Messud now. "No. But is it something that's more often true of women than men? I think so." She and her husband, the literary critic James Wood, have two children, a daughter, Livia, 12, and a son, Lucian, nine, and she goes on to tell me – with some nifty mimicry of the way children talk to one another – about the differences she observes in the kindergarten classes at their schools: "The boys go in the corner, they wave – Hi, Hi – and then they go off and build a Lego tower or whatever before school starts, and the girls go sit at a table and they draw and they look at each other's work and they chat and they say: 'Is that a flower? It doesn't look very much like a flower. Why did you use that colour? You used brown for a flower? Flowers aren't brown.'" It's very funny, until she adds the thought that: "The girl who doesn't sit at the table is a failed girl."

The Woman Upstairs is Messud's fifth book; each is strikingly different from the others. When the World Was Steady, the story of two sisters in the throes of mid-life crises of differing sorts, appeared in 1995, and was followed in 1999 by The Last Life, an ambitious story of four generations of a French-Algerian family (Messud herself is the daughter of a French-Algerian father and Canadian mother, and grew up in Australia, Canada and the US); Frank Kermode, tussling with the latter novel's complexities in a review, noted approvingly that "it was conceived as a work of art". Then came, in 2002, The Hunters, a brace of novellas intent on dissecting the very idea of the narrator; in the title piece, we have no idea of the name, age or gender of the person telling us their (somewhat spooky) story.

"The Hunters" is a bravura display of miniaturism and compression, in which every detail must be right for the effects to work; perhaps it was no accident that the other novella of the pair is called "A Simple Tale", an echo of Flaubert's "A Simple Heart", another work that condenses a servant's life into a perfectly turned short story. Neither piece prepared Messud's readership for her next book, The Emperor's Children (2006), a sparkling comedy of manners that features a trio of young New Yorkers beginning to make their way in the world of arts and letters before and after 9/11. It is a witty and sly look at the world of privilege and entitlement, but it also tackles the problem of "How to Live", the title of an unfinished manuscript by the novel's éminence grise.

Messud pauses for a long time when I ask her what, if anything, connects her books. She parries for a moment, joking that she has a terrible memory and can barely remember her first novel, but goes on to say that there is often a sense of displacement, and that she is interested in the places where we bump up against the limits of our interior narrators. She doesn't seem likely to go in for unnecessary mystification, so one suspects this is not coyness, but the result of an unwillingness or inability to see literature reduced to what's-it-about and technical process – to that shrunken head. At present, she teaches one semester a year in the master of fine arts programme at Hunter College in New York, and loves the enthusiasm her students bring, the "passionate, idiosyncratic engagement with the life of the mind"; one can imagine her as an inspiring presence in the seminar room, exacting yet generous. Recently her editor, Ursula Doyle, who has published all her books in this country, wrote a piece for Waterstones to accompany publication of The Woman Upstairs. "Every time we talk about a new book," Doyle writes, "it strikes me all over again how deeply and rigorously Claire has thought about exactly why she has done things as she has; her artistic vision is unwavering."

When she was in college, Messud remembers, she would do all the necessary groundwork and then she'd think: "I don't want to write it, the teacher doesn't want to read it, couldn't I just submit the notes and we'd be done?" "If you know what you're doing," she tells me, "it's not interesting. It has to be a challenge, it has to seem impossible and urgent to do it. And then you do it."